Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.     

Stress and failure analysis of a glass-epoxy composite plate with a circular hole

Experimental and finite-element analyses are presented for the anisotropic states of stress, strain and fracture of a glass-epoxy plate containing a circular hole and subjected to uniaxial tension. Strains were experimentally measured using foil gages, moiré and birefringent coating. Stresses are computed in the linear range from the measured strains. While the hole reduces the plate strength by a factor of two, the maximum tensile strain at fracture is greater than the ultimate strain in a plate without a hole. Fracture consists of crack initiation at the hole boundary but off the horizontal axis. Away from the hole, failure is accompanied by considerable delamination. Discontinuous crack propagation is present.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Stress state around cracks on the boundary of a hole in a photoelastic orthotropic plate under creep

The stress–strain state near cracks on the boundary of a circular hole in a linear elastic orthotropic composite plate under tension is analyzed. The distribution of stress intensity factors (SIFs) at the crack tip is found from photoelectric measurements. The dependence of the SIFs on the ratio of crack length to hole radius and on the mechanical properties of the material is established     

Bending of a highly stretched plate containing an eccentrically plate-reinforced circular hole

In this paper, we consider the problem of finding the stress distribution in a highly stretched plate containing a circular hole that is eccentrically reinforced by thickening the plate, on one side only, in an annular region concentric with the hole. A solution of the nonlinear Kármán plate equations is obtained that is asymptotically valid for large membrane stresses. We show that, except for a narrow bending boundary layer in the neighbourhood of the boundary between the reinforced area and the rest of the plate, a state of plane stress prevails and the reinforced area undergoes a transverse deflection that brings its middle surface into the plane of the middle surface of the plate.     

Elastoplastic analysis of a bent circular plate with a central hole

A circular aluminum plate with a small concentric hole (1/10 the plate thickness) and supported on its outer edge by a ring was subjected to a concentrated load at its center, applied through a rigid ball of radius equal to the plate thickness. Strains were determined using grids, moiré, and electrical strain gages on the top and bottom surfaces of the plate for loads up to and including the one associated with the appearance of the first crack in the plate. The investigation is related to the development of specimens to be used to determine fracture characteristics of materials used in lightweight construction.     

Circular welding problems with a crack

In this paper,the problem of an infinite plane with acircular hole welded by a nearly circular plate with a crackof different material is considered.The problem is tran-sformed into solving a certain boundary value problem ofanalytic functions and then reduced to solving a singularintegral equation along the crack.The formulas and somenumerical results of the factors of stress intensityfor the cases Mode Ⅰ and Mode Ⅱ are obtained at the endof this paper.     

Stress intensity factors of a plate with two cracks emanating from an arbitrary hole

In this paper, Muskhelishvili complex function theory and boundary collocation method are used to calculate the stress intensity factors (SIF) of a plate with two cracks emanating from an arbitrary hole. The calculated examples include a circular, elliptical, rectangular, or rhombic hole in a plate. The principle and procedure by the method is not only rather simple, but also has good accuracy. The SIF values calculated compare very favorably with the existing solutions. At the same time,the method can be used for different finite plate with two cracks emanating from a hole with more complex geometrical and loading conditions. It is an effective unified approach for this kind of fracture problems.     

Effect of friction in a contact problem for a plate with a pin

A solution is obtained for a contact problem concerning the tension of a rectangular elastic plate with a circular hole into which a rigid stationary pin has been inserted. There is a small gap between the hole and the pin, which is of circular cross section. Friction acts in the contact region in accordance with the Coulomb law. The finite-element method and the Boussinesq principle are used to determine the load that realizes a specified contact region. Two variants of boundary conditions on the contour of the hole are examined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 184–192, July–August, 1998.     

The solution of a crack emanating from an arbitrary hole by boundary collocation method

In this paper a group of stress functions has been proposed for the calculation of a crack emanating from a hole with different shape (including circular, elliptical, rectangular, or rhombic hole) by boundary collocation method. The calculation results show that they coincide very well with the existing solutions by other methods for a circular or elliptical hole with a crack in an infinite plate. At the smae time, a series of results for different holes in a finite plate has also been obtained in this paper. The proposed functions and calculation procedure can be used for a plate of a crack emanating from an arbitrary hole.     

Solution of a contact problem for a plate with a deformable insert

Contact problems with friction are solved for a rectangular plate with a circular hole into which a ring plate (insert) is placed with a small clearance. Two versions of contact boundary conditions are formulated. According to the proposed approximate formulation of the problem, the boundary conditions in both versions are satisfied not at the actual contact points but at specified pairs of points. Therefore, it is sufficient to determine attachment, slip, contact, and contact-free regions on just one of the contacting contours. The finite-element method and the Boussinesq principle are used to solve the problem. One of the versions of boundary conditions, compared to the other, gives smaller values for the strain energies of the plate and insert, the stress-concentration coefficient, and the lengths of attachment and contact regions. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 216–226, September–October, 1999.     

Stress distributions in a semi-infinite plate due to a pin determined by interferometric method

The stress distributions in a semi-infinite plate due to a loaded pin of the same material as the plate are systematically investigated by an interferometric method which has been developed by the authors. For the experiments, a finite plate of diallylphthalate with a circular hole is used. It is supported at one side and loaded in the direction normal to the opposing straight edge by a pin which just fits the hole. The ratio of the distancee between the hole center and the straight edge to the diameterd of the hole is varied in steps from 4.0 to 1.0. At each step, the distributions of principal stresses σ₁ and σ₂ along the hole edge, line of symmetry and straight edge, which have not been fully investigated especially whene/d is small, are obtained separately from the isopachic and isochromatic fringes of the interfero-stress patterns. The relations between the maximum values of these stresses and the shape factore/d are determined.     

Scattering of flexural wave in a thin plate with multiple circular holes by using the multipole Trefftz method

The scattering of flexural wave by multiple circular holes in an infinite thin plate is analytically solved by using the multipole Trefftz method. The dynamic moment concentration factor (DMCF) along the edge of circular holes is determined. Based on the addition theorem, the solution of the field represented by multiple coordinate systems centered at each circle can be transformed into one coordinate system centered at one circle, where the boundary conditions are given. In this way, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the scattering of flexural wave by multiple holes in an infinite plate subject to the incident flexural wave. The formulation is general and is easily applicable to dealing with the problem containing multiple circular holes. Although the number of hole is not limited in our proposed method, the numerical results of an infinite plate with three circular holes are presented in the truncated finite system. The effects of both incident wave number and the central distance among circular holes on the DMCF are investigated. Numerical results show that the DMCF of three holes is larger than that of one, when the space among holes is small and meanwhile the specified direction of incident wave is subjected to the plate.     

Creep and Stress Relaxation in a Plate Loaded Along the Contour of a Circular Hole

Analytical solutions of the creep and stress-relaxation boundary-value problems of a plate loaded externally along the contour of a circular hole are obtained using an unsteady creep model based on nonclassical representations for elastic and viscous properties of materials. It is assumed that one force component and one displacement component are specified at the boundary.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 146–153, July– August, 2005.     

Stress Concentration Factor in Functionally Graded Plates with Circular Holes Subjected to Anti-plane Shear Loading

Stress concentration factors due to the presence of geometrical discontinuities (circular holes) in functionally graded plates are derived. The material property inhomogeneity is assumed to be in the radial direction originating at the center of the plate. Variable separable closed-form solutions are obtained for the stresses and displacements in functionally graded plates (without and with holes) subjected to anti-plane shear loading. The stresses in functionally graded plates without a hole are not homogeneous as it is in the case of homogeneous plates. Either a stress concentration (more than the applied stress) or dilution (less than the applied stress) occurs depending on whether the modulus increases (hardening graded material) or decreases (softening graded material) away from the center of the graded plate without a hole. A novel definition of the stress concentration factor due to the geometrical discontinuity in functionally graded plates is derived. The effect of the circular hole in functionally graded plates is to magnify (compared to homogeneous plates) the stress concentration when the modulus decreases away from the center of the hole (softening material). Beneficial reduction of the stress concentration factor is achieved in hardening functionally graded materials.     

Fatigue growth modeling of cracks emanating from a circular hole in infinite plate

This paper presents a numerical approach of fatigue growth analysis of cracks emanating from a hole in infinite elastic plate subjected to remote loads. It involves a generation of Bueckner’s principle and a hybrid displacement discontinuity method (a boundary element method) proposed recently by the senior author of the paper. Because of an intrinsic feature of the boundary element method, a general crack growth problem can be solved in a single region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is modeled conveniently by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, fatigue growth process of an inclined crack in an infinite plate under uniaxial cycle load is modeled to illustrate the effectiveness of the numerical approach. In addition, fatigue growth of cracks emanating from a circular hole in infinite elastic plate subjected to remote loads is investigated by using the numerical approach. Many numerical results are given     

Singular integral equations for a patch repair problem

Within the scope of linear elasticity, an in-plane problem related to the repair of an infinite thin elastic plate with a hole by a patch is considered. The patch and the plate are joined together only along their boundaries. The plate is subjected to stresses applied at infinity. The problem is reduced to a system of four singular integral equations. Existence and uniqueness of the solution of the system is proved. The proposed solution allows one to evaluate the efficiency of a patch repair with little computational effort.     

Solution of the moiré hole drilling method using a finite-element-method-based approach

The moiré hole drilling method in a biaxially loaded infinite plate in plane stress is an inverse problem that exhibits a dual nature: the first problem results from first drilling the circular hole and then applying the biaxial loads, while the other problem arises from doing the opposite, i.e., first applying the biaxial load and then drilling the circular hole. The first problem is hardly ever addressed in the literature but implies that either separation of stresses or material property identification may be achieved from interpreting the moiré signature around the hole. The second is the well-known problem of determination of residual stresses from interpreting the moiré fringe orders around the hole. This paper addresses these inverse problem solutions using the finite element method as the means to model the plate with a hole, rather than the typical approach using the Kirsch solution, and a least-squares optimization approach to resolve for the quantities of interest. To test the viability of the proposed method three numerical simulations and one experimental result in a finite width plate are used to illustrate the techniques. The results are found to be in excellent agreement. The simulations employ noisy data to test the robustness of this approach. The finite-element-method-based inverse problem approach employed in this paper has the potential for use in applications where the specimen shape and boundary conditions do not conform to symmetric or well-used shapes. Also, it is a first step in testing similar procedures in three-dimensional samples to assess the residual stresses in materials.